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E-grāmata: Distributions, Partial Differential Equations, and Harmonic Analysis

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  • Formāts: PDF+DRM
  • Sērija : Universitext
  • Izdošanas datums: 20-Sep-2013
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781461482086
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Distributions, Partial Differential Equations, and Harmonic AnalysisPalielināt
  • Formāts: PDF+DRM
  • Sērija : Universitext
  • Izdošanas datums: 20-Sep-2013
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781461482086
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?The theory of distributions constitutes an essential tool in the study of partial differential equations. This textbook would offer, in a concise, largely self-contained form, a rapid introduction to the theory of distributions and its applications to partial differential equations, including computing fundamental solutions for the most basic differential operators: the Laplace, heat, wave, Lam\'e and Schrodinger operators.?

This book offers a rapid introduction to the theory of distributions and its applications to partial differential equations and harmonic analysis. Each chapter contains exercises and many have solutions at the end of the book.

Recenzijas

The book is very carefully written and its emphasis on applications in partial differential equations does justice to the nature and importance of the theory of distributions in modern analysis. An attractive feature of this work is the large number of exercises including solutions, both interspersed in the text and following each chapter. It is an ideal starting point to delve into the theory of distributions. (M. Kunzinger, Monatshefte für Mathematik, 2015)

The monograph under consideration aims at a thorough introduction to ideas and techniques of distribution theory with application to partial differential equations. the first part of this book is an excellent and carefully written introduction to the ideas and first applications of distribution theory which can be recommended for self-study for graduate students, while the second part contains a great variety of explicit formulas for tempered fundamental solutions interesting for the specialists in that area. (Michael Langenbruch, zbMATH 1308.46002, 2015)

The book under review deals with the distribution theory and its applications to partial differential equations. proofs are extremely detailed for the benefit of beginners, and there are a large number of exercises, more than half of which are accompanied by solutions. one appreciates the authors remarkable effort to present the subject in a form as clear and accessible as possible . This makes this textbook ideal for upper undergraduate to graduate level courses on the subject. (Fabio Nicola, Mathematical Reviews, September, 2014)

Preface vii
Introduction xiii
Common Notational Conventions xix
1 Weak Derivatives
1(16)
1.1 The Cauchy Problem for a Vibrating Infinite String
1(2)
1.2 Weak Derivatives
3(5)
1.3 The Spaces ε(Ω) and D(Ω)
8(5)
1.4 Additional Exercises for Chap. 1
13(4)
2 The Space D'(Ω) of Distributions
17(72)
2.1 The Definition of Distributions
17(8)
2.2 The Topological Vector Space D'(Ω)
25(2)
2.3 Multiplication of a Distribution with a C∞ Function
27(2)
2.4 Distributional Derivatives
29(5)
2.5 The Support of a Distribution
34(3)
2.6 Compactly Supported Distributions and the Space ε'(Ω)
37(11)
2.7 Tensor Product of Distributions
48(11)
2.8 The Convolution of Distributions in Rn
59(13)
2.9 Distributions with Higher-Order Gradients Continuous or Bounded
72(8)
2.10 Additional Exercises for Chap. 2
80(9)
3 The Schwartz Space and the Fourier Transform
89(20)
3.1 The Schwartz Space of Rapidly Decreasing Functions
89(10)
3.2 The Action of the Fourier Transform on the Schwartz Class
99(7)
3.3 Additional Exercises for Chap. 3
106(3)
4 The Space of Tempered Distributions
109(80)
4.1 Definition and Properties of Tempered Distributions
109(10)
4.2 The Fourier Transform Acting on Tempered Distributions
119(10)
4.3 Homogeneous Distributions
129(6)
4.4 Principal Value Tempered Distributions
135(6)
4.5 The Fourier Transform of Principal Value Distributions
141(5)
4.6 Tempered Distributions Associated with [ x]-n
146(5)
4.7 A General Jump-Formula in the Class of Tempered Distributions
151(11)
4.8 The Harmonic Poisson Kernel
162(5)
4.9 Singular Integral Operators
167(8)
4.10 Derivatives of Volume Potentials
175(9)
4.11 Additional Exercises for Chap. 4
184(5)
5 The Concept of a Fundamental Solution
189(12)
5.1 Constant Coefficient Linear Differential Operators
189(2)
5.2 A First Look at Fundamental Solutions
191(4)
5.3 The Malgrange-Ehrenpreis Theorem
195(5)
5.4 Additional Exercises for Chap. 5
200(1)
6 Hypoelliptic Operators
201(16)
6.1 Definition and Properties
201(2)
6.2 Hypoelliptic Operators with Constant Coefficients
203(7)
6.3 Integral Representation Formulas and Interior Estimates
210(6)
6.4 Additional Exercises for Chap. 6
216(1)
7 The Laplacian and Related Operators
217(60)
7.1 Fundamental Solutions for the Laplace Operator
217(6)
7.2 The Poisson Equation and Layer Potential Representation Formulas
223(10)
7.3 Fundamental Solutions for the Bi-Laplacian
233(4)
7.4 The Poisson Equation for the Bi-Laplacian
237(3)
7.5 Fundamental Solutions for the Poly-Harmonic Operator
240(8)
7.6 Fundamental Solutions for the Cauchy-Riemann Operator
248(5)
7.7 Fundamental Solutions for the Dirac Operator
253(7)
7.8 Fundamental Solutions for General Second-Order Operators
260(11)
7.9 Layer Potential Representation Formulas Revisited
271(3)
7.10 Additional Exercises for Chap. 7
274(3)
8 The Heat Operator and Related Versions
277(12)
8.1 Fundamental Solutions for the Heat Operator
277(3)
8.2 The Generalized Cauchy Problem for the Heat Operator
280(2)
8.3 Fundamental Solutions for General Second Order Parabolic Operators
282(5)
8.4 Fundamental Solution for the Schrodinger Operator
287(2)
9 The Wave Operator
289(20)
9.1 Fundamental Solution for the Wave Operator
289(18)
9.2 The Generalized Cauchy Problem for the Wave Operator
307(1)
9.3 Additional Exercises for Chap. 9
308(1)
10 The Lame and Stokes Operators
309(32)
10.1 General Remarks About Vector and Matrix Distributions
309(5)
10.2 Fundamental Solutions and Regularity for General Systems
314(5)
10.3 Fundamental Solutions for the Lame Operator
319(7)
10.4 Mean Value Formulas and Interior Estimates for the Lame Operator
326(6)
10.5 The Poisson Equation for the Lame Operator
332(2)
10.6 Fundamental Solutions for the Stokes Operator
334(4)
10.7 Additional Exercises for Chap. 10
338(3)
11 More on Fundamental Solutions for Systems
341(34)
11.1 Computing a Fundamental Solution for the Lame Operator
341(2)
11.2 Computing a Fundamental Solution for the Stokes Operator
343(1)
11.3 Fundamental Solutions for Higher-Order Systems
344(12)
11.4 Interior Estimates and Real-Analyticity for Null-Solutions of Systems
356(5)
11.5 Reverse Holder Estimates for Null-Solutions of Systems
361(4)
11.6 Layer Potentials and Jump Relations for Systems
365(8)
11.7 Additional Exercises for Chap. 11
373(2)
12 Solutions to Selected Exercises
375(40)
12.1 Solutions to Exercises from Sect. 1.4
375(6)
12.2 Solutions to Exercises from Sect. 2.10
381(16)
12.3 Solutions to Exercises from Sect. 3.3
397(3)
12.4 Solutions to Exercises from Sect. 4.11
400(7)
12.5 Solutions to Exercises from Sect. 5.4
407(1)
12.6 Solutions to Exercises from Sect. 6.4
408(1)
12.7 Solutions to Exercises from Sect. 7.10
409(4)
12.8 Solutions to Exercises from Sect. 9.3
413(1)
12.9 Solutions to Exercises from Sect. 10.7
413(1)
12.10 Solutions to Exercises from Sect. 11.7
414(1)
13 Appendix
415(34)
13.1 Summary of Topological and Functional Analytic
415(9)
13.2 Summary of Basic Results from Calculus, Measure Theory, and Topology
424(4)
13.3 Custom-Designing Smooth Cut-Off Functions
428(1)
13.4 Partition of Unity
429(5)
13.5 The Gamma and Beta Functions
434(1)
13.6 Surfaces in Rn and Surface Integrals
435(2)
13.7 Integration by Parts and Green's Formula
437(1)
13.8 Polar Coordinates and Integrals on Spheres
437(8)
13.9 Tables of Fourier Transforms
445(4)
Bibliography 449(6)
Subject Index 455(4)
Symbol Index 459
Dorina Mitrea is a Professor at University of Missouri in the Mathematics Department.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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